Are Christoffel symbols structure coefficients

Question

For a chart $(U,x^1,…,x^n)$ in an $n$ dimensional manifold, the the christoffel symbols for a covariant derivative are defined by $\nabla_{\partial_j}\partial_i=\Gamma_{ij}^k\partial_k$. For a general algebra of dimension $n$ with multiplication $\beta$, the structure constants are define by $\beta(x_i,x_j)=c_{ij}^kx_k$. Since the set of smooth vector fields is a vector space of $\mathbb{R}$, and the covariant derivative is bilinear, is this saying that the Christoffel symbols are exactly the structure constants of the algebra of smooth vector fields with the covariant derivative defined as multiplication? And if the connection is the Levi Cevita connection, is this saying the structure constants when multiplication is given by $\nabla$ is related to the structure constants when multplication is give by the lie bracket by $\Gamma_{ij}^k-\Gamma_{ji}^k=\gamma_{ij}^k$?

Answer 1

No, but they're related. If you study the Lie Group $G$ as a manifold, being $ \left\{ E_{1},...,E_{n}\right\}$ as a base of the tangent space in $p \in G$, you have as definition of Christoffel Symbols on $G$ $$\nabla_{E_{i}}E_{j}= {\sum}\Gamma_{ij}^{k}E_{k}$$ On the other side you have by definition of structure constants $$\left[E_{i},\,E_{j}\right]= {\sum}C_{ij}^{s}E_{s}.$$ Let's define the metric on the Lie Group from the adjoint action $$g\left(X,\,Y\right)=g\left(Ad_{p}\left(X\right),\,Ad_{p}\left(Y\right)\right) $$ Then using the metric $g$ to lower the index you have $$g\left(\left[E_{i},\,E_{j}\right],\,E_{k}\right)=C_{ijk},$$ Then you use the Koszul Formula to get the Levi-Civita connection $$\nabla_{E_{i}}E_{j}= {\sum}\frac{1}{2}\left(C_{kij}+C_{kji}+C_{ijk}\right)E_{k}.$$ Note that the relation with Christoffel Symbols is with structure constants with lowered indices that are the the structure constants involved with the metric. If you want a more clear picture you should have a look at the article of Milnor on Lie Groups and Curvature. Be aware of the sign convention, since I think he might use the opposite convention respect to Kobayashi Nomizu.